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It sounds like you broke the mechanical portion of the switch not the electronic portion.
. If you remove the lock cylinder you can use a flat screwdriver to start it but you will need to hold the key close to the hole because there is a micro chip inside it.
Here are the repair instructions You don't need the special tool a bent regular paper clip works, I have performed this repair many times it may take some time to get it to slide out.
not having any luck removing the cylinder. It won't come out. I put the key in the run position as best as I can determine and pushed the paper clip in the hole but the key keeps coming out after pulling on it for 10 seconds
You will need a small scredriver and pop off the ring around it first.